Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in PYTHON3):

Solution:
N = int(input())
P = list(map(float, input().split()))

# dp[i][j] := i番目までのコインを投げて、表がj枚出るものの確率
dp = [[0] * (N + 1) for i in range(N)]
dp[0][0] = (1 - P[0])
dp[0][1] = P[0]

for i in range(1, N):
    for j in range(i + 2):
        dp[i][j] = dp[i - 1][j] * (1 - P[i]) + dp[i - 1][j - 1] * P[i]

print(sum(dp[-1][N // 2 + 1:]))